1

To appear in the European Journal of Mechanics A/Solids

Digital Image Correlation used to Analyze the Multiaxial Behavior ofRubber-like Materials

by

L. Chevalier, S. Calloch, F. Hild and Y. Marco

LMT-Cachan

ENS de Cachan / CNRS – UMR 8535 / Université Paris 6

61 avenue du Président Wilson

94235 Cachan Cedex

France

2

Digital Image Correlation used to Analyze the Multiaxial Behavior ofRubber-like Materials

by L. Chevalier, S. Calloch, F. Hild and Y. Marco

Abstract

We present an experimental approach to discriminate models describing the mechanical

behavior of polymeric materials. A biaxial loading condition is obtained in a multiaxial

testing machine. An evaluation of the displacement field obtained by digital image correlation

allows us to evaluate the heterogeneous strain field observed during these tests. We focus on

the particular case of hyper-elastic models to simulate the behavior of a rubber-like material.

Different expressions of hyper-elastic potential are used to model experiments under uniaxial

and biaxial loading conditions.

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1. Introduction

The macromolecular structure of polymers leads to a specific behavior, in particular

large strain levels are generally observed. A large variety of models exists to simulate the

behavior of polymeric materials. These models give similar results for shear tests but lead to

different responses in extension for example. For the simple case of rubber-like materials at

room temperature, hyper-elasticity can be modeled in many different ways (Mooney 1940,

Rivlin 1948, Gent and Thomas 1958, Hart Smith 1966, Alexander 1968). The material

parameters are usually tuned in uniaxial tension but they give different responses for other

loading conditions (i.e., shear stress or multiaxial loading).

Multiaxial testing allows one to discriminate different models (see for example

Kawabata 1970, Meissner 1985). A specific testing machine called ASTREE enables us to

apply a multiaxial loading along three perpendicular directions. Consequently, strain field

heterogeneity may occur and does not lead to a simple identification. It follows that a

displacement field measurement is needed to check and quantify the heterogeneity of the

strain field. Optical measurement techniques are one of the most appealing methods to

estimate two-dimensional displacement fields. Upon loading, the observed surface moves and

deforms. By acquiring pictures for different load levels by using a Charge-Coupled Device

(CCD) camera, it is possible to determine the in-plane displacement field by matching

different zones of the two pictures. The simplest image-matching procedure is cross-

correlation that can be performed either in the physical space (Peters and Ranson 1981, Sutton

1983, Chu et al. 1985, Mguil et al. 1998, Sutton et al. 2000) or in the Fourier space (Chen et

al. 1993, Berthaud et al. 1996, Chiang et al. 1997, Collin et al. 1998). By using Fast Fourier

Transforms (FFT), one can evaluate the cross-correlation function very quickly.

The multiaxial testing machine used in this study is presented in Section 2. Section 3

introduces the basics related to the correlation of two signals. The correlation algorithm

(CORRELIGD) is presented. We then discuss the choice of the strain tensor when the

infinitesimal approximation is no longer valid. In the example of pure rotation, the precision

of the correlation technique is assessed by using an exact solution within the framework of

infinitesimal strains. Section 4 deals with an application using both ASTREE and

CORRELIGD tools. In particular, the choice of the size of the zone of interest in the

correlation technique is discussed when the displacements are not infinitesimal. An accurate

4

identification of the hyper-elastic potential is performed in case of a rubber-like material:

SmactaneTM.

2. A Multiaxial Testing Machine: ASTREE

The biaxial tests presented in this paper have been carried out on a triaxial testing

machine ASTREE (Fig. 1). This electro-hydraulic testing machine has six servohydraulic

actuators (Fig. 2). This machine has been developed by LMT-Cachan and Schenck AG,

Darmstadt, Germany. The load frame is composed of a fixed base, four vertical columns and a

mobile crosshead. The testing space 650 x 650 x 1500 mm3 in volume, delimited by the six

actuators, may be used for the tests. The two vertical actuators have a load capacity of 250 kN

and a stroke range of 250 mm. The four horizontal actuators have a load capacity of 100 kN

and a stroke range of 250 mm.

For efficient protection of the actuators at specimen failure (which can cause

extremely high side and twist forces), additional hydraulic bearings are installed in front of

each actuator. A hydraulic power station generates a flow of 330 l/min. Closed-loop control

for each actuator is provided by a digital controller, Schenck 59 serial hydropuls. The

controller monitors and provides signal conditioning for each load actuator position channel.

Each axis (X, Y and Z) has its own dedicated strain channel for signal conditioning and

control. Strain input signals can come from a variety of strain measuring devices (e.g., strain

gages, extensometer). Command waveform generation for each channel can be provided

either by the controller's internal waveform generators or externally via a personal computer.

Computer test control and data acquisition are performed by an object-oriented

programming software (LabVIEW®). The digital controller in combination with this software

package provides a highly versatile capability where numerous custom-made tests can be

developed. The digital controller allows each actuator to be driven independently or in

centroid control. A centroid mode uses a relationship between two opposite actuators along

the same axis to maintain the center motionless. Ensuring that the center does not move

prevents off-axis loading that would damage the actuators.

The digital controller enforces a centroid mode by using special algorithms to drive

the servocontrol of each actuator. Figure 2 illustrates how the centroid control algorithms

work for a displacement-controlled test. For each axis pair (e.g., Y+ and Y-), the controller

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uses the sum of the displacements and the difference of the displacements along the

considered axis. This centroid control mode has been used in all the multiaxial tests presented

hereafter.

3. Displacement field measurement by digital correlation

3.a Preliminaries: Correlation of Two Signals

To determine the displacement field of one image with respect to a reference image,

one considers a sub-image (i.e., a square region) which will be referred to as zone of interest

(ZOI). The aim of the correlation method is to match the ZOI in the two images (Fig. 3). The

displacement of one ZOI with respect to the other one is a two-dimensional shift of an

intensity signal digitized by a CCD camera. For the sake of simplicity, in the remainder of this

section the analysis is illustrated on one-dimensional signals. To estimate a shift between two

signals, one of the standard approaches utilizes a correlation function. One considers signals

g(ξ) which are merely perturbations of a shifted copy f(ξ−δ) of some reference signal f(ξ)

g(ξ) = f(ξ−δ) + b(ξ) (1)

where δ is an unknown displacement and b(ξ) a random noise. To evaluate the shift δ, one

may minimize the norm of the difference between f(ξ−∆) and g(ξ) with respect to ∆

min∆

||g - f(.−∆)||2 (2)

If one chooses the usual quadratic norm || f ||2 = ⌡⌠−∞+∞f(ξ)dξ, the previous minimization

problem is equivalent to maximizing the quantity h(∆)

h(∆) = (g ∗ f)(∆) = ⌡⌠−∞

+∞g(ξ)f(ξ−∆)dξ (3)

where ∗ denotes the cross-correlation product. Furthermore, when b is a white noise, the

previous estimate is optimal. The computation of a cross-correlation can be performed either

in the original space or in the Fourier space, by using an FFT

g ∗ f = N FFT−1 (FFT[g] FFT[f] ) (4)

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where the complex conjugate is overlined and N is the number of samples in the Fourier

transform (to use ‘fast’ algorithms, it is required that N = 2n, where n is an integer). The

previous results can be generalized to two-dimensional situations (i.e., image matching). The

use of the ‘shifting’ property enables one to ‘move’ a signal. Let us consider the shift operator

Td defined by

[Td f](ξ) = f(ξ−d) (5)

where d is the shift parameter. The FFT of Tdf becomes

FFT[ Td f] = Ed FFT[f] (6)

where the modulation operator Ed is defined by

[Ed f](ξ) = exp (-2π j d ξ) f(ξ) (7)

These two results constitute the basic tools for image correlation.

3.b Correlation Algorithm for Two-Dimensional Signals: CORRELIGD

Two images are considered. The first one, referred to as ‘reference image’ and the

second one, called ‘deformed image.’ One extracts the largest value p of a region of interest

(ROI) of size 2p x 2p pixel2 centered in the reference image. The same ROI is considered in

the deformed image. A first FFT correlation is performed to determine the average

displacement U 0 , V 0 of the deformed image with respect to the reference image. This

displacement is expressed in an integer number of pixels and is obtained as the maximum of

the cross-correlation function evaluated for each pixel of the ROI. This first prediction

enables the evaluation of the maximum number of pixels that belong to the two images. The

ROI in the deformed image is now centered at a point corresponding to the displaced center of

the ROI in the reference image by an amount U 0 , V 0 .

The user usually chooses the size of the zones of interest (ZOI) by setting the value of

s < p so that the size is 2s x 2s pixel2. To map the whole image, the second parameter to

choose is the shift δx (= δy) between two consecutive ZOIs: 1 ≤ δx ≤ 2s pixel. These two

parameters define the mesh formed by the centers of each ZOI used to describe the

displacement field. The following analysis is performed for each ZOI independently. It

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follows that parallel computations can be used in the present case. A first FFT correlation is

carried out and a first value of the in-plane displacement correction ∆U, ∆V is obtained. The

values ∆U, ∆V are again integer numbers so that the ZOI in the deformed image can be

displaced by an additional amount ∆U, ∆V. The absolute displacement residuals are now less

than 1/2 pixel in each direction. A sub-pixel iterative scheme can be used. To get good

localization properties of the Fourier transform, the considered ZOI is then windowed by a

modified Hanning window

ZOI = ZOI H ⊗ H (8)

where ZOI denotes the windowed ZOI, ⊗ the dyadic product and H the one-dimensional

modified Hanning window

H(i) =

12

1 − cos4πi

2s −1

when 0 < i < 2s −2

1 when 2s −2 < i < 3 × 2s −2

12

1− cos 4πi2 s −1

when 3 × 2 s−2 < i < 2 s −1

(9)

The parameter 2s−2 is an optimal value to minimize the error due to edge effects and to have a

sufficiently large number of data unaltered by the window (Hild et al. 1999). A cross-

correlation is performed. A sub-pixel correction of the displacement δU, δV is obtained by

determining the maximum of a parabolic interpolation of the correlation function. The

interpolation is performed by considering the maximum pixel and its eight nearest neighbors.

Therefore, one obtains a sub-pixel value. By using the ‘shifting’ property of the Fourier

transform, one can move the deformed ZOI by an amount δU, δV. Since an interpolation was

used, one may induce some errors requiring to re-iterate by considering the new ‘deformed’

ZOI until a convergence criterion is reached. The criterion checks whether the maximum of

the interpolated correlation function increases as the number of iterations increases.

Otherwise, the iteration scheme is stopped. The procedure, CORRELIGD (Hild et al., 1999), is

implemented in MatlabTM (1999). The precision of the method is at least of the order of 2/100

pixel and the minimum detectable displacement is also of the order of 1/100 pixel. For a

heterogeneous strain field, it is found that the precision on strain measurements is of the order

of 10−4 (Hild et al., 1999).

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3.c Finite Strain Measurements

For large displacements, two problems are to be solved: (i) the maximum

displacement related to the ZOI size; (ii) the choice of the strain measure.

When the same reference picture is used, it is not possible to measure large

displacements in a sequence of pictures. The solution is to follow the mesh by changing the

reference view when displacement becomes too big. When large displacements are supposed

to occur (i.e., in the present case) and a whole sequence of pictures is analyzed, the

displacement field is determined incrementally (i.e., by considering two consecutive images

of the sequence). This procedure can be simplified when the displacements remain small

enough (i.e., the first image remains the reference image throughout the whole analysis).

The first point will be further discussed by using a biaxial configuration presented

later on. The second question of this section is the choice of the strain tensor. It is well-known

that the infinitesimal strain tensor ε is not well adapted to large displacements for many

reasons. One of them is that there is no additivity. It follows that logarithmic strains are easier

to use from this point of view. Furthermore, for rigid body motion, the infinitesimal strains

can be different from zero in case of pure rigid displacements. Let us examine the case of a

pure rotation (see Fig. 4) by a 5° angle. In that case, the displacement field becomes:

( )xRu .1−= (10)

where R is an orthogonal second rank tensor, x the initial position with respect to the rotation

axis and 1 the unit second rank tensor. The infinitesimal strain tensor can be evaluated as:

( ) 0121 ≠−+= TRRε (11)

When the rotation is measured by one angle θ in the x-y plane, the non vanishing strain

components are εxx and εyy:

εxx = εyy = cosθ − 1 (12)

CORRELIGD gives an average value of both terms equal to –3.6.10-3 when s = 6 and δx = 32

and –4.10-3 when s = 7 and δx = 32. The exact value for θ = 5° is –3.8.10-3. The shear

component is equal to or less than 2.10-4. These results confirm that the precision of the strain

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measurements of the order of a 2.10-4 sensitivity. This incorporates the pixel interpolation

algorithm used to rotate images by using PhotoShop® (1998).

Despite this good result, it also confirms the necessity to use finite strain measures

when large rigid body motions are suspected to occur in addition to infinitesimal strains.

These measures are built by using the gradient F of the transformation and for the Lagrangian

measures, they can be expressed as:

( )( )

FFCmC

mCmE t

m

m . with0 when ln

21

0 when 121

=

=

≠−= (13)

where C denotes the right Cauchy-Green tensor. When m = 1 the Green-Lagrange tensor is

obtained, m=1/2 is the nominal strain tensor and yields ∆L/Lo for uniaxial elongation, m = 0 is

the logarithmic strain tensor; we will use this measure when not specified. By definition, all

these strains are equal to zero for a rigid rotation (F=R and C=1). The latter property can be

used to re-analyze the previous example of pure rotation. The average value of each

component of Green-Lagrange or logarithmic strain tensors is less than 2.10-4. This result is in

accordance with the previous sensitivity analysis in the framework of infinitesimal strains.

4. Study of a Rubber-Like Material

SmactaneTM is a rubber-like material developed by SMAC (Toulon – France). Various

stiffnesses and visco-elastic performances are proposed in sheet form or as final injected

product. This material was developed to obtain a maximum damping effect for a large range

of temperatures (i.e., from −50° to +120°C). One finds applications in the field of flexible

transmission joints, suspensions of machine, shock absorber. The grade used in this study

presents virtually no damping effect and its behavior is reversible over an elongation range of

800%. An ultimate strength of 9.5 MPa (Piola-Kirchhof stress) is given by the manufacturer.

During the tests, we will limit ourselves to lower stress levels to be sure that no other effect

than hyper-elasticity (e.g., irreversible effects, damage) occurs.

4.a Hyper-Elasticity of a Rubber-Like Material

The macroscopic approach of homogeneous, hyper-elastic media such as rubber-like

materials consists in the introduction of an elastic potential (Rivlin 1948). Classical

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hypotheses of isotropy, material-frame invariance and incompressibility allows one to assume

that the state potential W only depends on the first two invariant I1 and I2 of the right Cauchy-

Green tensor C

( ) 1=Cdet= with CtrCtr

21Ctr

321321

23

23

22

22

21

222

23

22

211

=

++=−=

++==λλλ

λλλλλλ

λλλI

I

I(14)

where λ1, λ2, λ3 are the three principal elongations. With such a potential, and by using the

conventional formalism of continuum thermodynamics, the hyper-elastic constitutive law

derives from W

∂∂−

∂∂=

∂∂=

∂∂ −

2

2

1122=

IWC

IW

CW

EWS (15)

where E is Green-Lagrange strain tensor and S the second Piola-Kirchhoff extra stress tensor.

The Cauchy extra stress tensor Σ is related to the Piola Kirchhoff extra stress tensor by

∂∂−

∂∂Σ −

2

1

12=..=

IWB

IWBFSF t (16)

We obtain the complete Cauchy stress tensor σ by using the relationship

Ip-= Σσ (17)

where p is the pressure associated with the incompressibility condition. Since the partial

derivatives of W with respect to I1 and I2 are known, the behavior is completely defined when

it remains hyper-elastic.

In this section, we discuss different potential expressions presented in Table 2. We can

see the forms proposed by Mooney (1940), Isihara (1951) and the neo-hookean (1941)

formalism are particular cases of the general Rivlin expression. This general form is also the

only one that leads to a first derivative ∂W/∂I1 that depends upon I1 and I2 as well as ∂W/∂I2.

It is worth noting that it may not be the case if we only consider the Cio and Coj terms of the

expansion. For all other forms, the expressions of the partial derivatives of W with respect to

I1 and I2 are uncoupled. As it is proposed by Lambert-Diani and Rey (1999) we call

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)( and )( 22

11

IgIWIf

IW =

∂∂=

∂∂ (18)

In the same paper, the authors proposed a strategy to identify both functions f and g by using

uniaxial and biaxial tests. Hyper-elastic models give the following Cauchy stress components

in uniaxial tension (σUT) and equibiaxial tension (σBT):

+=

+=

−−

−=

+=

+=

−−

−=

422

42

14

2242

1

22

21

222

1

2

12 with 1)(21)(2

21

2

with 1)(21)(2

λλ

λλ

λλλ

λσ

λλ

λλ

λλλ

λσ

I

IIgIf

I

IIgIf

BT

UT

(19)

Figure 5 shows that I1 is greater than I2 in uniaxial tension and that I2 is greater than I1 in

equibiaxial tension. We can also see that pure shear or plane strain tests lead to identical

values of I1 and I2. By using these results some authors (Heuillet 1997 for example) propose

to use such tests to identify the potential parameters. Since I1 is greater than I2 in uniaxial

tension, an approximate identification of the function f(I1) from uniaxial data is possible and

can be used to determine the function g(I2) from biaxial data (Lambert-Diani et al. 1999).

4.b Uniaxial Tension Test

A first tension test is carried out on SmactaneTM specimen by using a classical MTS

tension/compression testing machine. The tension test is carried out at room temperature (i.e.,

T = 22°C) and low strain rate (i.e., Ýε = 10-3 s-1). A typical result of displacement field

obtained by CORRELIGD is shown in Fig. 6 where one grip is fixed and the second one moves

from the right to the left on the picture. The displacements are large but the maximum strain is

only about 33%.

The problem now is that the initial mesh can move out of the picture so that for very

large strains we must choose a suitable initial mesh, evaluate the displacement step by step to

produce the tension curve, or move the CCD camera according to the previous displacement

value. By using ASTREE and the centroid mode enforces that the center of the specimen is

motionless. This point will be of extreme importance for biaxial tests (see Section 4.c).

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Experimental data obtained on a SmactaneTM specimen loaded in uniaxial tension are

shown in Fig. 7. Elongation is measured by using CORRELIGD and the average strain of the

ROI is obtained by using the CORRELIGAUGE procedure. The displacement field is first

interpolated by using cubic B-Spline functions. The strain field components are then derived

and the average values of the in-plane components are evaluated. The pictures of Fig. 6 are

related to the first four points of Fig. 7 from which we can evaluate the tangent Young’s

modulus E. This modulus is equal to 3.3 MPa.

If we assume that the contribution of the second invariant is negligible with respect to

the first one in a uniaxial stress σUT (since I1 >> I2), the function f can be directly identified

from the knowledge of the Cauchy stress and elongation λ

λλ

λλ

σ 2 with 12

)( 21

21+=

−

=ƒ II UT (20)

Figure 8 shows that this function is not constant during the tensile test on SmactaneTM. Apart

from the models proposed by Hart-Smith (1966) and Alexander (1968), all the others will not

be able to fit the experimental data. The two models lead to similar expressions of the first

invariant function

[ ] ( )2111 3ln)(ln −+= IkGIf (21)

where G and k1 are material parameters. To fit the experimental data obtained by Treolar

(1944) data, Lambert-Diani and Rey (1999) generalized this form

( )

−= ∑=

n

i

ii IaIf

011 3exp)( (22)

where ai are material parameters. These forms are compared in Fig. 8 for SmactaneTM. We

choose the value n = 2 because the experimental data are well described and the number of

parameters is minimal, even though the value n = 3 fits better the data. It is worth noting that

the form proposed by Rivlin assuming uncoupled evolution of f and g, would give as accurate

representation of the experimental data when choosing the same number of parameters. The

different coefficients are summarized in Table 3.

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The next question to address is the determination of the second invariant function

g(I2). By using a conventional uniaxial testing machine, a commonly used procedure is to

design large specimen (initial width Lo) with respect to the initial height ho (see Fig. 9). Since

the ratio h/L remains "small," a large area of the specimen is in a plane strain state (i.e.,

ε2 = 0). Edge effects (σ2 = 0) induce perturbations that propagate inside the specimen over a

length δ on both sides. Numerical simulations as well as video control on experimental "plane

strain" tests prove that δ is of the order of the specimen length h. In this zone the axial Cauchy

stress decreases from σ1 to κσ1 where κ value is about 0.6 for small strain levels.

If κ remains constant during the test, the error on stress measurement increases with

extension ratio λ and quickly reaches important values (i.e., up to 100%). The parameter κ

increases from 0.6 value to 1 when λ reaches about 5. The stress error decreases to zero (i.e.,

the stress distribution becomes homogeneous in the specimen) but we no longer are in plane

strain tension. The strain field is similar to a simple tension strain field. Besides, the invariants

I1 and I2 are equal during pure plane strain tension so that the effect of the function g is not

dominant for such a test. Consequently, a biaxial testing procedure is developed to identify

the second part of the hyper-elastic potential.

4.c Biaxial Tension Test

The material is equally stretched in both directions of the sheet (Fig.10). Fig. 10-b

shows typical CORRELIGD results obtained when a bad positioning of the specimen in the

grips occurs. The effect of rotation and elongation is superimposed with edge effects. This

picture is typical of asymmetric boundary conditions. CORRELIGD is of great help to validate

the positioning of the specimen. The specimen is slightly stretched and an analysis is run. It

takes at most a few minutes to get a full displacement field and to check its homogeneity. It

follows that the experimental boundary conditions can be checked a priori and used a

posteriori to simulate the experiment.

It is easy to understand that the first evaluation of the displacement vector obtained by

correlation of the ZOI will not be accurate as soon as the effective displacement is greater

than half the length of the ZOI. In Fig. 11a we can observe that the evaluation of the

displacement field is not satisfactory. The biaxial extension of the plane specimen is about 7%

in both directions. With a 32 pixel ZOI (i.e., s = 5), only a small square region leads to

reasonable results of the correlation technique in agreement with the previous remark (the

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range of the displacement contours is equal to 40 pixel). This zone increases with a 64 pixel

ZOI (the displacement range is about 2/3 of the size of the ZOI). Some artifacts remain in the

lower part of Fig. 11b. The solution is to choose a larger ZOI (s = 7) when the displacement

becomes important. Figure 11c shows that the problem is solved and that the displacement

field is linear in both directions. Table 1 shows the average strains over the whole ROI for the

three sizes of ZOI. When s = 5, the average longitudinal strains are negative even though the

specimen was stretched. This first result clearly shows again that s = 5 is too small for this

displacement and strain range. The value s = 6 leads to reasonable results. However, one may

note that the strains are different in the two directions. When s = 7 the shear strain is less than

precision of the correlation technique. The longitudinal strains are virtually identical in the

two directions: this experiment is indeed representative of an equibiaxial stretching. It is

worth noting the very good homogeneity of the strain field with such a simple set-up. We will

see that this homogeneity is also observed even with larger displacements.

Figures 12a, 12b and 12c show the displacement field after biaxial testing on a

SmactaneTM specimen. The corresponding strain field is not completely homogeneous (Fig.

12d). By considering the strain quasi-constant along AB (see Fig. 12a and 12c), we assume

that the stresses are uniform in this section AB. Consequently, we can estimate the Cauchy

stress (i.e., σBT = 2 F/eL where e is the current specimen thickness and L the current length

between A and B). The result is plotted in Fig. 13 from which we can deduce the initial slope.

This slope value is equal to 7.6 MPa that is less than 10% different from the theoretical value

‘2E’ for uncompressible material.

The biaxial Cauchy stress helps us to identify the form of the second invariant function

g(I2) by using Eqn. (23)

24

242

1

24

42

2 and 12 with 12

1)(2)(

1

2 λλ

λλ

λλ

λλσ

+=+=

−

−ƒ−= II

IIg

TB(23)

Figure 14 shows the shape of the function g(I2) which can be fitted by the models proposed by

Gent and Thomas (1958), Hart-Smith (1966) or Alexander (1968). Two different forms are

analyzed in this figure: (i) the Hart-Smith form; (ii) the Lambert-Diani and Rey form,

respectively

15

∑ == ni i

iIcIg 02

2)( (24)

( )

−= ∑ =

ni

ii IbIg 0 22 3exp)( (25)

But the identification does not stop here; CORRELIGD evaluates the whole strain field at each

step of the test and by using the estimated material coefficients we can calculate the stress

field. It follows that the forces in each direction can be computed and compared with

measured load. This method leads to optimize material parameters for the chosen constitutive

law. By inverse analysis of the problem, we modify these coefficients to fit as well as possible

the experimental data. In Fig. 15, the simulations using the Lambert-Diani and Rey model are

compared with the uniaxial data. In the same figure, the simulation using only the f function

contribution is plotted. A posteriori, one can observe the small influence of the second

invariant function g, on the uniaxial simulation since experimental data are well fitted with the

complete model (i.e., f and g contributions.)

5. Conclusions and Perspectives

A multiaxial testing procedure has been used to analyze the behavior of a rubber-like

material. By using a CCD camera coupled with digital image correlation, we can measure the

displacement field over all the specimen. This approach is particularly well-adapted for large

displacement tests where it is difficult to obtain homogeneous strain fields. Such a non

intrusive method helps one to analyze more thoroughly experiments on rubber-like as well as

other materials. In particular, the experimental boundary conditions can be checked.

Uniaxial and equibiaxial tests have been carried out on SmactaneTM. The results are

analyzed by assuming that the contribution of the invariants I1 and I2 are uncoupled in the

hyper-elastic potential. Experimental data are used to discriminate classical hyper-elastic

models. For a SmactaneTM material, the Mooney-Rivlin model is not representative of the

hyper-elastic behavior when strains over 100% occur. The models proposed by Alexander or

Hart-Smith are more representative even though the time-dependent part of the behavior was

not studied. By using such models, we can simulate other loading conditions (e.g., plane

strain, biaxial tension) and compare deformed shapes and strain fields with numerical

simulations of the test. This constitutes one of the perspectives to this work. Another

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perspective is to take into account the strain rate effects (by comparing successive strain field

measurements for small time intervals). Such an analysis can be used to analyze the visco-

elastic properties of rubber-like materials and other polymers.

Acknowledgments

The authors wish to thank Dr. Jean-Noël Périé who has been of great help in the

CORRELI programming under the Matlab5.3 environment. SMAC-Toulon provided us with

SmactaneTM sheets.

17

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19

List of tables

Table 1: Average in-plane strains for three different sizes of ZOI.

Table 2: Hyper-elastic potential expressions.

Table 3: Material parameters for the function f when n = 1, 2 & 3 (SmactaneTM).

Table 4: Material parameters for the function g when n = 1 & 2 (SmactaneTM).

List of figures

Figure 1: Multiaxial testing machine ASTREE.

Figure 2: Coupled actuators for the multiaxial testing machine ASTREE.

Figure 3: ZOI in the initial and ‘deformed’ image.

Figure 4: Displacement contours in pure rotation (s = 6 and δx = 32 pixels).

Figure 5: Invariants I1 and I2 for different tests.

Figure 6: Large displacements measured with CORRELIGD in a uniaxial tensile test on

SmactaneTM.

Figure 7: Stress/strain response of SmactaneTM in uniaxial tension. Initial stress/strain curve in

axial and equibiaxial tension.

Figure 8: Identification of the first invariant function for different values of n.

Figure 9: 'Plane strain' test, principle and limitation.

20

Figure 10: Positioning of the specimen during biaxial testing in the multiaxial machine

ASTREE.

Figure 11: Influence of ZOI size on the evaluation of the displacement field in a biaxial test.

Figure 12: Biaxial stress measurement, displacement and strain field in a biaxial test on

SmactaneTM.

Figure 13: Stress/strain response of SmactaneTM in uniaxial and equibiaxial tension.

Figure 14: Second invariant function identification.

Figure 15: Description of the hyper-elastic behavior of SmactaneTM.

21

εxx (%) εyy (%) εxy (%)

s = 5 -0.10 -0.40 0.09

s = 6 5.50 4.50 0.37

s = 7 5.47 5.42 <0.01

Table 1: Average in-plane strains for three different sizes of ZOI.

22

Author(s) Hyper-elastic potential Number of

material

parameters

Mooney (1940) W = C1(I1-3) + C2(I2-3) 2

Neo-Hookeen (1943) W = G/2 (I1-3) 1

Rivlin (1943) W = Cij(I1-3)i(I2-3)j ?

Isihara (1951) W = C1(I1-3) + C2(I2-3)2 + C3(I2-3) 3

Rivlin, Saunders (1951) W = C1(I1-3) + f(I2-3) with: f(0)=0 ?

Gent and Thomas (1958) W = C1(I1-3) + C2 ln(I2/3) 2

Biderman (1958) W = C10(I1-3) + C20(I1-3)2 + C30(I1-3)3 + C01(I2-3) 4

Hart-Smith (1966)W as: ( )

2

2

2

211

1 and )3(exp

IkG

IWIkG

IW =

∂∂−=

∂∂ 3

Alexander (1968) W= ( )

−+

+−+−∫ )3(3ln)3(exp2 23

221

211 ICICdIIkC

γγµ 5

Table 2: Hyper-elastic potential expressions.

23

n = 1 n = 2 n = 3 n = 4

a0 = -1.38

a1 = -1.03 10-2

a0 = -1.27

a1 = -2.52 10-2

a2 = 1.40 10-3

a0 = -1.86

a1 = -8.01 10-2

a2 = 7.10 10-3

a3 = -1.45 10-4

a0 = -1.14

a1 = -1.31 10-1

a2 = 1.65 10-2

a3 = -7.00 10-4

a4 = 1.02 10-5

Table 3: Material parameters for the function f when n = 1, 2 3 & 4 (SmactaneTM).

(i) n = 1 (i) n = 2 (ii) n = 1 (ii) n = 2 (ii) n = 3

c0 = 3.25 10-1

c1 = 2.76 10-1

c0 = -9.53 10-3

c1 = 2.89

c2 = -4.39

b0 = -1.00

b1 = -0.01

b0 = -1.02

b1 = -3.92 10-3

b2 = -5.30 10-4

b0 = -1.03

b1 = -7.22 10-3

b2 = -2.05 10-3

b3 = -1.70 10-4

Table 4: Material parameters for the function g (SmactaneTM).

(i) Hart-Smith model when n = 1 & 2

(ii) Lambert-Diani and Rey model when n = 1, 2 & 3

24

Figure 1: Multiaxial testing machine ASTREE.

25

Figure 2: Coupled actuators for the multiaxial testing machine ASTREE.

26

Figure 3: ZOI in the initial and ‘deformed’ image.

27

Figure 4: Displacement contours in pure rotation (s = 6 and δx = 32 pixels).

28

Figure 5: Invariants I1 and I2 for different tests.

29

Figure 6: Large displacements measured with CORRELIGD in a uniaxial tensile test on

SmactaneTM.

30

Figure 7: Stress/strain response of SmactaneTM in uniaxial tension. Initial stress/strain curve in

axial and equibiaxial tension.

31

Figure 8: Identification of the first invariant function for different values of n.

32

Figure 9: 'Plane strain' test, principle and limitation.

33

Figure 10: Positioning of the specimen during biaxial testing in the multiaxial machine

ASTREE.

34

Figure 11: Influence of ZOI size on the evaluation of the displacement field in a biaxial test.

35

Figure 12: Biaxial stress measurement, displacement and strain field in a biaxial test on

SmactaneTM.

36

Figure 13: Stress/strain response of SmactaneTM in uniaxial and equibiaxial tension.

37

Figure 14: Second invariant function identification.

38

Figure 15: Description of the hyper-elastic behavior of SmactaneTM.